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In hypothesis testing, a commonly used starting point is establishing the null hypothesis. This is the presumption that there is no effect or difference in the situation we are examining. When considering a population's characteristics, the null hypothesis often reflects the status quo or a specific claim about a population parameter.
In the context of our exercise, the null hypothesis is expressed as follows:

• \( H_0: p = 0.24 \) - meaning that the proportion of college students who prefer the color blue is the same as the general population, which is 24%.

The null hypothesis acts as the basis for all statistical inferences. It is the hypothesis we seek evidence against when performing a statistical test. If our data provide enough evidence against it, we may reject the null hypothesis. By structuring our assumptions this way, we can determine whether there is a significant difference present based on our collected data.

On the flip side of the null hypothesis, we have the alternative hypothesis. This hypothesis suggests that there is indeed a difference or effect and is what researchers are usually trying to support.
In the exercise example, our alternative hypothesis is stated as:

• \( H_a: p eq 0.24 \) - indicating that the proportion of college students who prefer blue is different from 24% (either greater or lesser).

The alternative hypothesis is essential as it defines the direction of the test. In our example, we are conducting a two-tailed test, which checks for any significant deviation (either increase or decrease) in the percentage compared to the general population.
Researchers use the formulation of the alternative hypothesis to focus their analysis on detecting specific changes they are interested in exploring. If statistical evidence supports the alternative hypothesis, it suggests that the status quo (as described by the null hypothesis) may not hold true.

The one-sample z-test is a statistical method used to assess if the sample mean differs significantly from a known population mean. When dealing with proportions, similar principles apply.
For our exercise, the one-sample z-test for proportions is utilized to compare the sample proportion (\( \hat{p}\)) from college students with the general population proportion (\( p_0 \)). The formula used is:

• \( z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \)

Key Steps to perform a One-Sample Z-Test for Proportions:

• Calculate the sample proportion, \( \hat{p} \), by dividing the number of successes by the sample size, \( n \).
• Substitute \( \hat{p} \), \( p_0 \), and \( n \) into the z-test formula to determine the z-score.
• Compare the computed z-value with the critical z-value(s) from standard normal distribution tables (e.g., \( \pm 1.96 \) for a 5% significance level in a two-tailed test).

If the computed z-value falls beyond the critical values, the null hypothesis is rejected. If it lies within, like in our case (\( z = -0.426 \)), it suggests insufficient evidence to claim a significant difference in the population proportions.